Download On some questions about b

On some questions about b-open sets
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M. Ganster and M. Steiner
Abstract
In [2], D. Andrijevic observed that for every topological space (X, τ ) we have
P O(X, τ ) ∪ SO(X, τ ) ⊆ BO(X, τ ) ⊆ SP O(X, τ ), but that none of these inclusions
can be replaced by equality. The purpose of this note is to characterize those spaces
(X, τ ) where BO(X, τ ) = SP O(X, τ ) holds, and those spaces (X, τ ) where P O(X, τ ) ∪
SO(X, τ ) = BO(X, τ ) holds.
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Introduction and Preliminaries
Over the years quite a number of generalizations of the class of open sets in a topological
space have been considered and widely investigated. These variations have many useful
applications, e.g. they are utilized to provide results about decompositions of continuity.
Among the most important classes of generalized open sets are the following:
Definition 1 A subset S of a topological space (X, τ ) is called
(i) semi-open [9], if S ⊆ intS ,
(ii) preopen [10], if S ⊆ intS ,
(iii) semi-preopen [3] or β-open [1], if S ⊆ intS ,
(iv) b-open [2], if S ⊆ intS ∪ intS .
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2000 Math. Subject Classification — Primary: 54A10;
Key words and phrases — b-open, preopen, semi-open, semi-preopen, resolvable, strongly irresolvable, extremally disconnected.
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The classes of semi-open, preopen, semi-preopen and b-open subsets of a space (X, τ ) are
usually denoted by SO(X, τ ), P O(X, τ ), SP O(X, τ ) and BO(X, τ ) respectively. The class
of b-open sets has been introduced and investigated in 1996 by Andrijevic [2]. He made the
following fundamental observation.
Proposition 1.1 For every space (X, τ ), P O(X, τ ) ∪ SO(X, τ ) ⊆ BO(X, τ ) ⊆ SP O(X, τ )
holds but none of these implications can be reversed.
The aim of this paper is to provide complete answers to the following open questions.
Question 1. For which spaces (X, τ ) does BO(X, τ ) = SP O(X, τ ) hold?
Question 2. For which spaces (X, τ ) does P O(X, τ ) ∪ SO(X, τ ) = BO(X, τ ) hold?
Recall that a space (X, τ ) is said to be resolvable if it has two disjoint dense subsets,
otherwise it is called irresolvable. Every space (X, τ ) can be represented uniquely as the
disjoint union of a closed and resolvable subspace F and an open, hereditarily irresolvable
subspace G (see e.g. [7] and [5]). We shall call this decomposition the Hewitt-representation
of (X, τ ). A space (X, τ ) is called strongly irresolvable if every open subspace of (X, τ ) is
irresolvable. Clearly, if X = F ∪ G denotes the Hewitt-representation of (X, τ ) then (X, τ )
is strongly irresolvable if and only if G = X .
A space (X, τ ) is called extremally disconnected if U is open for every open subset U of
(X, τ ). Following E. van Douwen [4], x ∈ X is said to be an e.d.-point of (X, τ ) if x ∈ U
implies x ∈ intU for every open subset U of (X, τ ). We shall denote the set of e.d.-points of
a space (X, τ ) by ED(X, τ ). Clearly, a space (X, τ ) is extremally disconnected if and only
if ED(X, τ ) = X.
For the convenience of the reader we now list some known results that will be used later
in our paper.
Proposition 1.2 Let (X, τ ) be a space.
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(i) If U ⊆ X is open and S ⊆ X is semi-open (preopen, semi-preopen, b-open respectively) then U ∩ S is semi-open (preopen, semi-preopen, b-open respectively),
(ii) for every subset A ⊆ X, A ∩ intA is preopen,
(iii) [2] S ⊆ X is b-open if and only if S is the union of a semi-open set and a preopen
set.
Proposition 1.3 For a space (X, τ ) the following are equivalent:
(1) (X, τ ) is extremally disconnected,
(2) SO(X, τ ) ⊆ P O(X, τ ),
(3) SP O(X, τ ) = P O(X, τ ),
(4) BO(X, τ ) = P O(X, τ ).
Proof.
(1) ⇔ (2) has been shown in [8] and (1) ⇔ (3) has been shown in [6]. Clearly,
(3) ⇒ (4) and (4) ⇒ (2) follow immediately from Proposition 1.1.
Proposition 1.4 For a space (X, τ ) the following are equivalent:
(1) (X, τ ) is strongly irresolvable,
(2) P O(X, τ ) ⊆ SO(X, τ ),
(3) SP O(X, τ ) = SO(X, τ )
(4) BO(X, τ ) = SO(X, τ ).
Proof.
(1) ⇔ (2) has been shown in [5] and (1) ⇔ (3) has been shown in [6]. Clearly,
(3) ⇒ (4) and (4) ⇒ (2) follow immediately from Proposition 1.1.
Proposition 1.5 [6] For a space (X, τ ) the following are equivalent:
(1) SP O(X, τ ) = SO(X, τ ) ∪ P O(X, τ ),
(2) (X, τ ) is extremally disconnected or strongly irresolvable.
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The results
We start by characterizing those spaces where the class of b-open sets coincides with the
class of semi-preopen sets. The Hewitt-representation of a space (X, τ ) will be denoted by
X =F ∪G .
Theorem 2.1 For a space (X, τ ) the following are equivalent:
(1) BO(X, τ ) = SP O(X, τ ),
(2) W is open for every open resolvable subspace W ,
(3) G is open and intF ⊆ ED(X, τ ),
(4) (X, τ ) is the topological sum of an extremally disconnected space and a strongly
irresolvable space.
Proof.
(1) ⇒ (2):
If W = ∅, we are done, so let W be a nonempty open resolvable
subspace and let E1 and E2 be disjoint dense subsets of W . Pick x ∈ W . If x ∈ W
then clearly x ∈ intW . If x ∈
/ W , let S = E1 ∪ {x}. It is easily checked that S = W ,
hence intS = W and so S ∈ SP O(X, τ ). If U is an open set such that U ⊆ S, then
∅ = U ∩ E2 = U ∩ E2 = U ∩ W = U . Thus intS = ∅. By hypothesis, S ∈ BO(X, τ ) and so
x ∈ S ⊆ intS ∪ intS = intS = intW . Thus W is open.
(2) ⇒ (3): Since intF is an open resolvable subspace, intF is open by hypothesis. Since
F is closed and intF ⊆ F , we have intF = intF and so G = intG, i.e. G is open. Now
let x ∈ intF and let U be open such that x ∈ U . If W = U ∩ intF then W is open and
resolvable and x ∈ W . By hypothesis, x ∈ intW ⊆ intU and so x is an e.d.-point of (X, τ ).
(3) ⇒ (4): Clearly, X = intF ∪ G and G is a clopen strongly irresolvable subspace and intF
is clopen and extremally disconnected as a subspace.
(4) ⇒ (1): Let X = E ∪ C be the topological sum of an extremally disconnected subspace E
and a strongly irresolvable subspace C. Let S ∈ SP O(X, τ ). Since E and C are clopen, it is
easily verified that S∩E ∈ SP O(E, τ |E ) , hence S∩E ∈ P O(E, τ |E ) by Proposition 1.3 and
so S∩E ∈ P O(X, τ ). In a similar manner, S∩C ∈ SP O(C, τ |C ) , hence S∩C ∈ SO(C, τ |C )
by Proposition 1.4 and so S ∩ C ∈ SO(X, τ ). Since S = (S ∩ E) ∪ (S ∩ C) , S is the union
of a preopen set and a semi-open set and thus b-open, i.e. S ∈ BO(X, τ ). 2
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Corollary 2.2 Let (X, τ ) be resolvable. Then BO(X, τ ) = SP O(X, τ ) if and only if (X, τ )
is extremally disconnected.
In our next result we characterize those spaces where the class of b-open sets coincides
with P O(X, τ ) ∪ SO(X, τ ).
Theorem 2.3 For a space (X, τ ) the following are equivalent:
(1) P O(X, τ ) ∪ SO(X, τ ) = BO(X, τ ),
(2) for each subset A ⊆ X , intA ⊆ intA or intA ⊆ intA.
Proof.
(1) ⇒ (2):
Let A ⊆ X and let B = (A ∩ intA) ∪ intA . It is easily checked
that intA = intB and that intA = intB (see e.g. [6], Proposition 11). Since A ∩ intA ∈
P O(X, τ ) and intA ∈ SO(X, τ ) , we have that B ∈ BO(X, τ ). By hypothesis, B ∈
P O(X, τ ) ∪ SO(X, τ ) . If B ∈ P O(X, τ ), then intA ⊆ B ⊆ intB = intA. If B ∈ SO(X, τ ),
then A ∩ intA ⊆ B ⊆ intB = intA and consequently intA ⊆ intA = A ∩ intA ⊆ intA .
(2) ⇒ (1): Let S ∈ BO(X, τ ). If intS ⊆ intS , then S ⊆ intS and so S ∈ SO(X, τ ). If
intS ⊆ intS , then S ⊆ intS and so S ∈ P O(X, τ ). 2
As a consequence we have the following new characterization of spaces (X, τ ) where the
class of semi-preopen sets coincides with P O(X, τ ) ∪ SO(X, τ ) (concerning other characterizations we refer the reader to [6]).
Corollary 2.4 For a space (X, τ ) the following are equivalent:
(1) P O(X, τ ) ∪ SO(X, τ ) = SP O(X, τ ),
(2) W is open for every open resolvable subspace W , and for each subset A ⊆ X ,
intA ⊆ intA or intA ⊆ intA.
It is obvious that the condition
P O(X, τ ) ∪ SO(X, τ ) = SP O(X, τ ) implies that
P O(X, τ ) ∪ SO(X, τ ) = BO(X, τ ). In our next example we point out that this implication cannot be reversed in general.
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Example 2.5 Let X1 and X2 be disjoint infinite sets, let p ∈
/ X1 ∪ X2 and let X =
X1 ∪ X2 ∪ {p} . A topology τ on X is defined in the following way: a basic neighbourhood
of x ∈ Xi is a cofinite subset of Xi , i ∈ {1, 2} , a basic neighbourhood of p has the form
{p} ∪ C1 ∪ C2 where Ci is a cofinite subset of Xi , i ∈ {1, 2} .
Clearly (X, τ ) is a T1 space and X1 and X2 are open subspaces of (X, τ ). If Di and Ei
are disjoint infinite subsets of Xi , i ∈ {1, 2} , then D1 ∪ D2 and E1 ∪ E2 are disjoint dense
subsets of (X, τ ), and so (X, τ ) is resolvable. Since p ∈ X1 \ intX1 , (X, τ ) is not extremally
disconnected and so SP O(X, τ ) 6= P O(X, τ ) ∪ SO(X, τ ) by Proposition 1.5.
Now let A ⊆ X. If A is finite, then A is closed and so intA = intA ⊆ intA . Hence, we
assume that A is infinite. If both A ∩ X1 and A ∩ X2 are infinite, then A is dense and so
intA ⊆ intA . So we may assume without loss of generality the case that A ∩ X1 is infinite
and A ∩ X2 is finite. Then p ∈
/ intA , A ∩ X1 = X1 ∪ {p} and A ∩ X2 = A ∩ X2 . If A ∩ X1
is not cofinite in X1 , then intA = ∅ and so intA ⊆ intA . If A ∩ X1 is cofinite in X1 , then
intA = A ∩ X1 and intA = X1 ∪ {p} . Furthermore, A = (X1 ∪ {p}) ∪ (A ∩ X2 ) and
intA = X1 , thus intA ⊆ intA .
Hence, for every subset A ⊆ X we have intA ⊆ intA or intA ⊆ intA , i.e. BO(X, τ ) =
P O(X, τ ) ∪ SO(X, τ ) by Theorem 2.3. 2
We conclude our paper with a result that for a large class of spaces, however, the conditions ”P O(X, τ ) ∪ SO(X, τ ) = SP O(X, τ )” and ”P O(X, τ ) ∪ SO(X, τ ) = BO(X, τ )”
coincide. Recall that a space (X, τ ) is called quasi-regular [11] if every nonempty open set
contains a nonempty regular closed set (i.e. the closure of a nonempty open set).
Theorem 2.6 For a quasi-regular space (X, τ ) the following are equivalent:
(1) P O(X, τ ) ∪ SO(X, τ ) = SP O(X, τ ),
(2) P O(X, τ ) ∪ SO(X, τ ) = BO(X, τ ).
Proof. (1) ⇒ (2): This is obvious.
(2) ⇒ (1): If (X, τ ) is strongly irresolvable, then SP O(X, τ ) = P O(X, τ ) ∪ SO(X, τ ) by
Proposition 1.5. So we assume that (X, τ ) has a nonempty open resolvable subspace W . Let
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E1 and E2 be disjoint dense subsets of W . We claim that (X, τ ) is extremally disconnected.
Suppose that (X, τ ) is not extremally disconnected. Then there is a point x ∈ X and an
open set U such that x ∈ U \ intU = U ∩ X \ U . Then W ∩ (U ∪ (X \ U ) 6= ∅ , and without
loss of generality we assume that W ∩ (X \ U ) = W1 6= ∅ . Since (X, τ ) is quasi-regular,
there is a nonempty open set V such that V ⊆ W1 . Let S = U ∪ {x} . Then S is semi-open
but not preopen. Let T = V ∩ E1 . Then T is preopen but not semi-open, since intT = ∅ .
We obviously have S ⊆ U and T ⊆ X \ U , and T ⊆ V and S ⊆ X \ V .
If we set A = S ∪ T , then clearly A is b-open. Hence, by assumption, A is preopen or
semi-open. If A is preopen, then A ∩ (X \ V ) = S is preopen, a contradiction. If A is
semi-open, then A ∩ (X \ U ) = T is semi-open, a contradiction. Thus (X, τ ) is extremally
disconnected and so SP O(X, τ ) = P O(X, τ ) ∪ SO(X, τ ) by Proposition 1.5. 2
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[11] J. C. Oxtoby, Spaces that admit a category measure , J. Reine Angew. Math. 205
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Maximilian GANSTER, Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-98010 Graz, Austria.
Markus STEINER, Department of Mathematics, Graz University of Technology, Steyrergasse
30, A-98010 Graz, Austria.
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